DISCLAIMER: I'm primarily a software guy, for the most part of my professional life I've been a software developer, and while I have a non-negligible amount of knowledge about electronics, this has never been my area of expertise, not by a long shot.
I do realize this is a very odd question to ask, since things don't seem to work like this nowadays in the industry, but this is my very first time questioning myself about things like these, and I had to start somewhere - here's hoping this is the right place to start.
Is it ever possible for a guy like me, provided I learn the required process to do it (as well as having the money for it), to eventually ask Intel to create a chip based entirely on my own design, using some kind of VHDL language, or anything related to it?
I'm specifically excluding FPGAs, ASICs, or any solutions designed to create custom logic with their own logic boards. Do I have the possibility to, eventually, hold a 500+-pin chip (say, a CPU) in my own hands after asking Intel to follow my own design and create a chip out of it?
"Ask Intel to create a chip ..."
You can ask all you want. You will be ignored unless you show up with a check in hand for, say, US$100M engineering cost and a commitment to buy US$1000M parts over say 10 years time.
Why exclude FPGAs? You can do your own full custom processor design today and fit it in a 500+/-pin FPGA and cost way less than the above option. You can easily do it on a US$500 high end FPGA development board.
The NRE cost today of doing a full custom design is well beyond the limit of any hobbyists, except maybe Elon Musk or another hobbyist billionaire.